A car can slow down at 5.10 m/s2 without skidding when coming to rest on a level road. What would its acceleration be if the road were inclined at 12o uphill?
To determine the car's acceleration when the road is inclined at 12° uphill, we need to find the component of gravity parallel to the incline.
The force due to gravity can be split into two components: one perpendicular to the incline and one parallel to the incline. The component parallel to the incline is given by:
F_parallel = mg * sinθ
where m is the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s²), and θ is the angle of incline (12°).
Since the car is slowing down without skidding, the frictional force (F_friction) opposing the motion of the car is equal to the net force. The frictional force is given by:
F_friction = m * a
where a is the acceleration of the car.
Setting these two forces equal to each other, we have:
F_parallel = F_friction
mg * sinθ = m * a
Simplifying, we can cancel out the mass from both sides of the equation:
g * sinθ = a
Substituting the values, we get:
a = 9.8 m/s² * sin(12°)
Using a calculator, we find:
a ≈ 2.09 m/s²
Therefore, the car's acceleration would be approximately 2.09 m/s² when the road is inclined at 12° uphill.
To find the car's acceleration when the road is inclined, we need to consider the forces acting on the car. These forces include the force due to gravity and the force opposing the motion (friction).
First, let's find the force due to gravity. The force due to gravity can be calculated using the formula:
Force due to gravity = mass * gravitational acceleration
The gravitational acceleration is a constant value which is approximately equal to 9.8 m/s². However, since the car is moving uphill, we also need to consider the component of gravitational acceleration along the incline.
The component of gravitational acceleration along the incline can be calculated using the formula:
Component of gravitational acceleration along the incline = gravitational acceleration * sin(θ)
where θ is the angle of the incline (12° in this case).
Next, let's consider the force opposing the motion (friction). When the car is moving uphill, the frictional force acts downhill, opposite to the direction of the car's motion. The magnitude of the frictional force can be calculated using the formula:
Frictional force = mass * acceleration
In this case, the frictional force is equal to the car's mass multiplied by its acceleration.
Now, we can set up the equation of motion to find the car's acceleration. The equation is:
Force due to gravity - Frictional force = mass * acceleration
Substituting the values we calculated earlier:
(mass * gravitational acceleration * sin(θ)) - (mass * acceleration) = mass * acceleration
Let's rearrange the equation to solve for acceleration:
(mass * gravitational acceleration * sin(θ)) = 2 * mass * acceleration
Now, we can cancel out the mass on both sides of the equation:
gravitational acceleration * sin(θ) = 2 * acceleration
Finally, we can solve for acceleration:
acceleration = (gravitational acceleration * sin(θ)) / 2
Substituting the values for gravitational acceleration (9.8 m/s²) and the angle of the incline (12°):
acceleration = (9.8 m/s² * sin(12°)) / 2
Using a scientific calculator to evaluate sin(12°) and calculating the final expression, we find:
acceleration ≈ 0.212 m/s²
Therefore, the car's acceleration when the road is inclined at 12° uphill is approximately 0.212 m/s².